Fig. REF  shows different kinds of transposition graphs corresponding to the generating sets \(S_1,\,S_2,\ldots ,\,S_6\)  for some \(n\)  and \(k\) . In this paper a transposition graph having \(n\)  vertices are labeled from 1 to \(n\)  without any duplication. The Cayley graph generated by \(S_1\)  is called the bubble-sort graph \(\text{BS}_n\)  [1]}, [2]}, by \(S_2\)  is called the hypercube graph \(\text{HC}_n\)  [3]}, by \(S_3\)  is called the star graph \(\text{ST}_n\)  [1]}, [2]}, [3]}, by \(S_4\)  is called the complete transposition graph \(\text{CT}_n\)  [2]}, by \(S_5\)  is called the generalized star graph \(\text{GST}_{n, k}\)  [3]}, and by \(S_6\)  is called the modified bubble sort graph \(\text{MBS}_n\)  [2]}.
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